TY - JOUR

T1 - Study of the stability for three-dimensional states of dynamic equilibrium of the electron Vlasov-Poisson gas

AU - Gubarev, Yuriy G.

AU - Liu, Yang

N1 - This work was supported partially by China Scholarship Council.

PY - 2023

Y1 - 2023

N2 - Here, in this paper, the spatial motions of a boundless collisionless electron Vlasov-Poisson gas were considered in a three-dimensional Cartesian coordinate system. Using the replacing of independent variables in the form of hydrodynamic substitution, a transition was made from the kinetic equations to an infinite system of gas-dynamic equations in the "vortex shallow water"and the Boussinesq approximations. In the process of proving the linear instability for the exact stationary solutions to the Vlasov-Poisson equations, the well-known sufficient Newcomb-Gardner-Rosenbluth condition for stability of these solutions with respect to one incomplete unclosed particular class of small spatial perturbations was reversed. An original linear differential second-order inequality with constant coefficients for the Lyapunov functional was also obtained. When the sufficient conditions found in this paper for the linear practical instability of the exact stationary solutions under research are satisfied, an a priori exponential estimate from below for the growth of small three-dimensional perturbations follows from this inequality. Since the estimate was derived without any additional restrictions on the exact stationary solutions under study, then, in this way, the absolute linear instability for the spatial dynamic equilibrium states of the electron Vlasov-Poisson gas with respect to three-dimensional perturbations was proved.

AB - Here, in this paper, the spatial motions of a boundless collisionless electron Vlasov-Poisson gas were considered in a three-dimensional Cartesian coordinate system. Using the replacing of independent variables in the form of hydrodynamic substitution, a transition was made from the kinetic equations to an infinite system of gas-dynamic equations in the "vortex shallow water"and the Boussinesq approximations. In the process of proving the linear instability for the exact stationary solutions to the Vlasov-Poisson equations, the well-known sufficient Newcomb-Gardner-Rosenbluth condition for stability of these solutions with respect to one incomplete unclosed particular class of small spatial perturbations was reversed. An original linear differential second-order inequality with constant coefficients for the Lyapunov functional was also obtained. When the sufficient conditions found in this paper for the linear practical instability of the exact stationary solutions under research are satisfied, an a priori exponential estimate from below for the growth of small three-dimensional perturbations follows from this inequality. Since the estimate was derived without any additional restrictions on the exact stationary solutions under study, then, in this way, the absolute linear instability for the spatial dynamic equilibrium states of the electron Vlasov-Poisson gas with respect to three-dimensional perturbations was proved.

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85176790009&origin=inward&txGid=380b3510c9a5146b251844b203b6b64e

UR - https://www.mendeley.com/catalogue/8286070b-4076-3875-a117-1852e437a43a/

U2 - 10.1063/5.0163155

DO - 10.1063/5.0163155

M3 - Conference article

VL - 2872

JO - AIP Conference Proceedings

JF - AIP Conference Proceedings

SN - 0094-243X

IS - 1

M1 - 060024-1

ER -